When an `alpha-`particle of mass 'm' moving with velocity 'v' bombards on a heavy nucleus of charge 'Ze' its distance of closest approach from the nucleus depends on `m` as :

Options

A. m
B. `1/m`
C. `1/sqrtm`
D. `1/m^(2)`

Answer

To find the distance of closest approach of an alpha particle to a heavy nucleus, we can equate the kinetic energy of the alpha particle to the electrostatic potential energy at the point of closest approach. Here’s a step-by-step solution:

### Step 1: Understand the scenario
An alpha particle with mass \( m \) and charge \( +2e \) is moving towards a heavy nucleus with charge \( +Ze \). As it approaches the nucleus, it experiences a repulsive electrostatic force due to the positive charges.

### Step 2: Write down the kinetic energy of the alpha particle
The kinetic energy (KE) of the alpha particle is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the alpha particle and \( v \) is its velocity.

### Step 3: Write down the electrostatic potential energy at the closest approach
The electrostatic potential energy (PE) between the alpha particle and the nucleus at a distance \( r_0 \) (the distance of closest approach) is given by:
\[ PE = \frac{k \cdot (Ze) \cdot (2e)}{r_0} \]
where \( k \) is Coulomb's constant.

### Step 4: Set the kinetic energy equal to the potential energy
At the distance of closest approach, the kinetic energy of the alpha particle is equal to the electrostatic potential energy:
\[ \frac{1}{2} mv^2 = \frac{k \cdot (Ze) \cdot (2e)}{r_0} \]

### Step 5: Rearrange the equation to find \( r_0 \)
Rearranging the equation to solve for \( r_0 \):
\[ r_0 = \frac{2k \cdot Ze \cdot 2e}{mv^2} \]
This simplifies to:
\[ r_0 = \frac{4kZe^2}{mv^2} \]

### Step 6: Analyze the dependency on mass \( m \)
From the expression for \( r_0 \), we can see that the distance of closest approach \( r_0 \) is inversely proportional to the mass \( m \):
\[ r_0 \propto \frac{1}{m} \]

### Conclusion
Thus, the distance of closest approach of the alpha particle to the nucleus depends inversely on its mass.

### Final Answer
The distance of closest approach \( r_0 \) is inversely proportional to the mass \( m \) of the alpha particle.

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